1. Field of the Invention
The invention relates to oil well production. More particularly, the invention relates to methods for optimizing oil well production.
2. State of the Art
The crude oil which has accumulated in subterranean reservoirs is recovered or xe2x80x9cproducedxe2x80x9d through one or more wells drilled into the reservoir. Initial production of the crude oil is accomplished by xe2x80x9cprimary recoveryxe2x80x9d techniques wherein only the natural forces present in the reservoir are utilized to produce the oil. However upon depletion of these natural forces and the termination of primary recovery, a large portion of the crude oil remains trapped within the reservoir. Also many reservoirs lack sufficient natural forces to be produced by primary methods from the very beginning. Recognition of these facts has led to the development and use of many enhanced oil recovery techniques. Most of these techniques involve injection of at least one fluid into the reservoir to force oil towards and into a production well.
Typically, one or more production wells will be driven by several injector wells arranged in a pattern around the production well(s). Water is injected through the injector wells in order to force oil in the xe2x80x9cpay zonexe2x80x9d of the reservoir towards and up through the production well. It is important that the water be injected carefully so that it forces the oil toward the production well but does not prematurely reach the production well before all or most of the oil has been produced. Generally, once water reaches the production well, production stops. Over the years, many have attempted to calculate the optimal pumping rates for injector wells and production wells in order to extract the most oil from a reservoir.
An oil reservoir can be characterized locally using well logs and more globally using seismic data. However, there is considerable uncertainty as to its detailed description in terms of geometry and geological parameters (e.g. porosity, rock permeabilities, etc.). In addition, the market value of oil can vary dramatically and so financial factors may be important in determining how production should proceed in order to obtain the maximum value from the reservoir.
As early as 1958, a linear programming model was proposed by Lee, A. S. and Aronovsky, J. S. in xe2x80x9cA Linear Programming Model for Scheduling Crude Oil Production,xe2x80x9d J. Pet. Tech. Trans. A.I.M.E. 213, pp. 51-54. More recently, in 1974, the optimum number and placement of wells has been calculated using mixed integer programming. See, Rosenwald, G. W. and Green, D. W., xe2x80x9cA Method for Determining the Optimum Location of Wells in a Reservoir Using Mixed Integer Programming,xe2x80x9d Society of Petroleum Engineers of AIME Journal, Vol. 14, No. 1, February 1974, p 44-54. In the 1980s work was done regarding the optimum injection policy for surfactants. This work maximized the difference between gross revenue and the cost of chemicals in a one-dimensional situation but with a sophisticated set of equations simulating multiphase flow in a porous medium. See, Fathi, Z. and Ramirez, W. F., xe2x80x9cUse of Optimal Control Theory for Computing Optimal Injection Policies for Enhanced Oil Recovery,xe2x80x9d Automatica 22, pp. 33-42 (1984) and Ramirez, W. F., xe2x80x9cApplications of Optimal Control Theory to Enhanced Oil Recovery,xe2x80x9d Elsevier, Amsterdam (1987). Most recently, in the 1990s, the Pontryagin Maximum Principle for Autonomous Time Optimal Control Problems and Constrained Controls has been applied to optimize oil recovery. See, Sudaryanto, B., xe2x80x9cOptimization of Displacement Efficiency of Oil Recovery in Porous Media Using Optimal Control Theory,xe2x80x9d Ph.D. Dissertation, University of Southern California, Los Angeles (1998) and Sudaryanto, B. and Yortsos, Y. C., xe2x80x9cOptimization of Displacement Efficiency Using Optimal Control Theoryxe2x80x9d, European Conference on the Mathematics of Oil Recovery, Peebles, Scotland (1998). Because of the linear dependence of the Hamiltonian on the control variables, if the variables are constrained to lie between upper and lower bounds, the Pontryagin Maximum Principle implies that optimal controls display a xe2x80x9cbangxe2x80x94bang behaviorxe2x80x9d, i.e. each control variable staying at one bound or the other. This leads to an efficient algorithm.
All of these approaches to optimizing oil recovery are subject to various uncertainties. Some of these uncertainties include the accuracy of the mathematical model used, the accuracy and completeness of the data, financial market fluctuations, the possibility that new information will affect present measurements, and the possibility that new technology will affect the collection and/or interpretation of data. Choosing a course of action will invariably involve some risk.
It is therefore an object of the invention to provide methods for optimizing oil recovery from an oil reservoir.
It is also an object of the invention to provide methods for optimizing oil recovery from an oil reservoir which takes into account both deterministic and stochastic factors.
It is another object of the invention to provide methods for optimizing oil recovery from an oil reservoir which account for downside risk.
It is still another object of the invention to provide methods for optimizing oil recovery from an oil reservoir which takes into account both financial as well as physical parameters.
In accord with these objects which will be discussed in detail below, the methods of the present invention include the application of portfolio management theory to associate levels of risk with Net Present Values (NPV) of the amount of oil expected to be extracted from the reservoir. Using the methods of the invention, production parameters such as pumping rates can be chosen to maximize NPV without exceeding a given level of risk, or, for a given level of risk, the NPV can be maximized with a 90% confidence level.
More particularly, the methods of the invention include first deriving semi-analytical results for a model of the reservoir. This involves setting up a forward problem and the corresponding deterministic problem. Certain simplifying assumptions are made regarding viscosity, permeability, the oil-water interface, the initial areal extent of the oil, the shape of the oil patch and its location relative to the production well. With these assumptions, the motion of the oil-water interface is derived under the influence of oil production at a central well and water injection at neighboring wells. The flow rates (pumping rates) are constrained by positive lower and upper bounds determined by the well and formation structures. The amount of oil extracted, or its NPV is optimized under the assumption that production stops when water breaks through at the producer well. According to the methods of the invention, flow rates do not change continuously. A time interval is split into a small number of subintervals during which flow rates are constant. Optimizing flow rates according to the invention is an optimization of a function of several variables (the flow rates in all the time intervals) rather than a classical control problem contemplated by the Pontryagin Maximum Principle. The solution exhibits a xe2x80x9cbang bang behaviorxe2x80x9d with each control variable staying mainly at one bound or the other.
After considering this deterministic problem, a probabilistic description is created by assuming that the precise areal extent of the remaining oil is not known. An uncertainty such as this is affected by one or more numerical parameters which are referred to herein as uncertainty parameters. By appropriate averaging over multiple realizations, forming expectations by numerical integration, the expected NPV is maximized for a set of flow rates and a risk aversion constant. The probability distribution of the NPV and its uncertainty (i.e. the variance given the values of the control variables which optimize the mean) are also calculated. The results are then represented as probability distribution curves for the NPV and for total production (given that the flow rates are chosen to optimize the expected NPV). The probability distributions of the financial outcomes can then be calculated from the probability distributions describing the uncertain reservoir parameters. Efficient frontiers (similar to those described in Markowitz""s theory of portfolio management) are then calculated by optimizing the linear combinations of the expected NPV and its standard (or semi-) deviation. Each point on the efficient frontier corresponds to a set of flow rates which will produce a maximum expected NPV with a given risk.
An iterative process for carrying out the invention includes the following steps.
(a) Choose a risk aversion constant K.
(b) Choose a set of flow rates.
(c) For each of certain chosen values of the uncertainty parameters, calculate and store an objective function (e.g. NPV).
(d) Calculate the mean and variance of the objective function set obtained in step (c) to obtain an objective function FK of the risk aversion constant, FK being a linear combination of semi-variance and mean NPV.
(e) repeat steps (b) through (d) until an optimal FK is found for the risk aversion constant K,
(f) when the optimal FK is found for the risk aversion constant K, store the means and variances calculated in step (d),
(g) repeat steps (a) through (f) for each risk aversion constant, and
(h) generate an efficient frontier based on the set of means and variances stored in step (f).
Additional objects and advantages of the invention will become apparent to those skilled in the art upon reference to the detailed description taken in conjunction with the provided figures.